73.25.22 problem 35.4 (h)

Internal problem ID [15730]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (h)
Date solved : Tuesday, January 28, 2025 at 08:06:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+\left (-2 x^{3}+5 x \right ) y^{\prime }+\left (-x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 31

Order:=6; 
dsolve(2*x^2*diff(y(x),x$2)+(5*x-2*x^3)*diff(y(x),x)+(1-x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1-\frac {1}{6} x^{2}-\frac {1}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 34

AsymptoticDSolveValue[2*x^2*D[y[x],{x,2}]+(5*x-2*x^3)*D[y[x],x]+(1-x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_2 \left (-\frac {x^4}{56}-\frac {x^2}{6}+1\right )}{x}+\frac {c_1}{\sqrt {x}} \]